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Seminari de Geometria Algebraica 2018/2019 |
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Conferenciant
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Títol |
Data i hora |
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ARCADES Doctoral School II and ESR Days
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3 a 7 de setembre, IMUB
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Constantin Shramov
Steklov Math Inst & NRU HSE
Moscou, Rússia
Contact:
ignasi.mundet at ub.edu
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Automorphisms of Kaehler manifolds
I will survey various results about finite groups acting by automorphisms and birational
automorphisms of Kaehler manifolds. I will show that in many cases such groups enjoy the Jordan property, similar to subgroups
of general linear groups.
The talk is based on joint works with Yu. Prokhorov.
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Martín Sombra
ICREA - UB
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The zero set of the independence polynomial of a graph
In statistical mechanics, the independence polynomial of a graph G arises as the partition
function of the hardcore lattice gas model on G.
The distribution of the zeros of these polynomials when G→∞ is relevant for
the study of this model and, in particular, to the determination of its phase transitions.
In this talk, I will review the known results on the location of these zeros,
with emphasis on the case of rooted regular trees of fixed degree and varying depth k ≥ 0.
Our main result states that for these graphs, the zero sets of their independence
polynomials converge as k→∞ to the bifurcation measure,
in the sense of DeMarco, of a certain family of dynamical systems on the Riemann sphere.
This is ongoing work with Juan Rivera-Letelier (Rochester)
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Divendres 28 de setembre, 15h, Aula T2, FMI-UB
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Alberto F. Boix
Ben-Gurion U. of the Negev
Beer-Sheva, Israel
Contact:
szarzuela at ub.edu
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A Characteristic Free Approach to Finite Determinacy
Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real and complex-analytic category and in the differentiable
case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given
by a group action and the first step is always to reduce the determinacy question to an “infinitesimal determinacy”, i.e. to the tangent spaces at the orbits of the group action.
The goal of this talk is to formulate a universal approach to finite determinacy in arbitrary characteristic, not necessarily over a field, for a large class of group
actions; along the way, we introduce the notion of “pairs of (weak) Lie type”, which are groups together with a substitute for the tangent space at the unit element,
such that the group is locally approximated by its tangent space, in a precise sense. This construction may be regarded as a sort of replacement of the
exponential/logarithmic maps and is of independent interest. In this generality we establish the “determinacy versus infinitesimal determinacy” criteria, a far reaching
generalization of numerous classical and recent results, together with some new applications.
The content of this talk is based on joint work with Gert–Martin Greuel (Universität Kaiserslautern, Germany) and Dmitry Kerner (Ben–Gurion University of
the Negev, Israel)
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Divendres 5 d'octubre, 15h, Aula T2, FMI-UB
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Roberto Gualdi
U. Bordeaux - UB - CRM
Contact:
sombra at ub.edu
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Height of cycles in toric varieties
We present in this talk some relations between suitable heights of cycles
in toric varieties and the combinatorics of the defining Laurent polynomials.
To do this, we associate to any Laurent polynomial f with coefficients in an
adelic field two families of concave functions on a certain real vector space:
the upper functions and the Ronkin functions of f.
For the choice of an adelic semipositive toric metrized divisor D, we give
upper bounds for the D-height of a complete intersection in a toric variety
in terms of the upper functions of the defining Laurent polynomials.
In the one-codimensional case, we prove an exact formula relating the D-height of a hypersurface to the Ronkin function of the associated Laurent
polynomial, generalizing the well-known equality for the canonical case.
Our approach involves mixed integrals, Legendre-Fenchel duality and other
notions from convex geometry.
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Divendres 19 d'octubre, 15h, Aula T2, FMI-UB
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Francisco Presas
ICMAT, Madrid
Contact:
ignasi.mundet at ub.edu
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Homotopy type of the space of smooth embeddings of \(\Large{\mathbb{S}}^1\) in \(\Large{\mathbb{R}}^4\) via Engel geometry.
We introduce the space of horizontal embeddings for the standard Engel distribution in the Euclidean 4-space.
We prove that the space of smooth embeddings of the circle into R⁴ is simply connected
(classical result), by checking that the space of horizontal embeddings has homotopy type very related to the space of smooth embeddings (they are related by an h-principle).
We extend the method to sketch the computation of the \(\pi_2\) of that space showing that is Z\(\oplus\)Z (more modern result).
We finally comment on work in progress further generalizing the techniques by using Manifold calculus to try to compute the whole homotopy type of this space.
This is joint work with E. Fernández and X. Martínez-Aguinaga.
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Divendres 26 d'octubre, 15h, Aula T2, FMI-UB
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Pedro D. González Pérez
ICMAT - UCM, Madrid
Contact:
adefelipe at ub.edu
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The valuative tree is a projective limit of Eggers-Wall trees
Consider a germ C of reduced curve on a smooth germ S of complex analytic surface.
Assume that C contains a smooth branch L.
Using the Newton-Puiseux series of C relative to any coordinate system (x,y) on S such that L is the y-axis,
one may define the Eggers-Wall tree ΘL(C) of C relative to L.
Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders.
The main objective of this paper is to embed canonically ΘL(C) into Favre and
Jonsson's valuative tree of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural
functions on ΘL(C) as pullbacks of other naturally defined functions on the valuative tree.
As a consequence, we prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning one branch:
if L' is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees
ΘL'(C) and ΘL(C) identify them canonically,
their associated triples of functions being easily expressible in terms of each other.
We prove also that the valuative tree is the projective limit of Eggers-Wall trees over all choices
of curves C.
Joint work with Evelia García Barroso and Patrick Popescu-Pampu.
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Divendres 9 de novembre, 15h, Aula T2, FMI-UB
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Joana Cirici
UB
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A Dolbeault cohomology theory for almost complex manifolds
In this talk I will survey recent joint work with Scott Wilson which extends Dolbeault cohomology to all almost complex manifolds,
and generalizes some foundational results for compact Kähler manifolds to the non-integrable setting.
I will also explain Lie-algebra analogues of the theory which provide useful computational tools for compact Lie groups and nilmanifolds.
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Divendres 16 de novembre, 15h, Aula T2, FMI-UB
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Diletta Martinelli
School of Mathematics
Edinburgh, Escòcia (UK)
Contact:
marti.lahoz at ub.edu
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On the geometry of contractions of the Moduli Space of sheaves of a K3 surface
I will describe how recent advances have made possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland.
In particular Bayer and Macrì relate birational transformations of the moduli space M of sheaves of a K3 surface X to wall-crossing in the space of Bridgeland stability conditions Stab(X).
I will explain how it is possible to refine their analysis to give a precise description of the geometry
of the exceptional locus of any birational contractions
of M.
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Divendres 23 de novembre, 15h, Aula T2, FMI-UB
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Bernd Sturmfels
Max Plank Institut Leipzig, Alemanya
Contact:
cdandrea at ub.edu
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Moment Varieties of Polytopes
The moments of the uniform distribution on a convex polytope are rational
functions in its vertex coordinates. We study the projective varieties
parametrized by these moments. This is work with Kathlen Kohn and Boris
Shapiro. On our journey, we encounter Hankel determinantal ideals, splines,
cumulants, multisymmetric functions, and invariants of nonreductive groups.
While moment varieties are complicated, they offer many nice open problems.
Article
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Dilluns 26 de novembre, 15h, aula T1, FMI-UB
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Vincenzo Antonelli
Politecnico di Torino, Itàlia
Contact:
miro at ub.edu
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Ulrich bundles on Hirzebruch surfaces
Ulrich bundles on a projective variety are vector bundles
that admit a completely linear resolution as sheaves on
the projective space. They carry many interesting
properties and they are the simplest one from the
cohomological point of view.
In this talk we characterize Ulrich bundles of any rank on
polarized rational ruled surfaces over P¹. We show that
every Ulrich bundle admits a resolution in terms of line
bundles. Conversely, given an injective map between
suitable totally decomposed vector bundles, we show that
its cokernel is Ulrich if it satisfies a vanishing in
cohomology.
Finally we discuss some particular cases and we construct
examples of indecomposable Ulrich bundles.
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Divendres 30 de novembre, 15h, Aula T2, FMI-UB
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Enrico Carlini
Politecnico di Torino,
Itàlia
Contact:
alessandro.oneto at upc.edu
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Hilbert function of double points
Hilbert functions of zero dimensional schemes, reduced or not, play a crucial role in many areas of mathematics: from Waring ranks of forms to identifiability of tensors. However, while we have a very good understanding of the reduced case, we know very little in the not reduced case. In this talk we will explore the situation with a special focus to double points in the plane.
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Divendres 30 de novembre, 16h, Aula T2, FMI-UB
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FACARD 2019 Workshop
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16 a 18 de gener, IMUB
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Laura Brustenga
UAB
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Relative clusters for smooth families
In the talk, we will discuss a generalisation of clusters of points to the relative setting.
When the family is smooth, we are able to show that relative clusters form a representable functor.
We will recall the construction of Kleiman's iterated blow-ups, which are the representing schemes for the absolute case.
Thereafter we will focus on and work out an explicit example of length two relative clusters. The example is geometric and interesting in its own, but hopefully, it will also share some insight about the general situation.
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Divendres 1 de febrer, 15h, Aula T2, FMI-UB
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Elba Garcia-Failde
IPHT (CNRS) París-Saclay, França
Contact:
carles.casacuberta at ub.edu
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Simple maps, topological recursion and a new ELSV formula
In this talk, we call ordinary maps a certain type of graphs embedded on surfaces, in contrast to fully simple maps,
which we introduce as maps with non-intersecting disjoint boundaries.
It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR).
We propose a combinatorial interpretation of the important and still mysterious symplectic transformation which
exchanges x and y in the initial data of the TR (the spectral curve).
We give elegant formulas for the disk and cylinder topologies which recover relations already known in the context of free probability.
For genus zero we provide an enumerative geometric interpretation of the so-called higher order free cumulants,
which suggests the possibility of a general theory of approximate higher order free cumulants taking into account the higher genus amplitudes.
We also give a universal relation between fully simple and ordinary maps through double monotone Hurwitz numbers,
which can be proved either using matrix models or bijective combinatorics.
In particular, we obtain an ELSV-like formula for double strictly monotone Hurwitz numbers with ramification profile
(2,...,2) over 0 and arbitrary one over ∞.
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Divendres 8 de febrer, 15h, Aula T2, FMI-UB
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Marco Gualtieri
University of Toronto, Canadà
Contact:
eva.miranda at upc.edu
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The potential of generalized Kahler geometry
Since the introduction of generalized Kähler geometry in 1984 by
Gates, Hull, and Roček in the context of two-dimensional supersymmetric
sigma models, we have lacked a general understanding of the degrees of
freedom inherent in the geometry. In particular, the description of a usual
Kähler structure in terms of a complex manifold together with a local
Kähler potential function is not available for generalized Kähler
structures, despite many positive indications in the literature over
the last decade. I will explain how holomorphic Poisson geometry may
be used to solve this problem and to obtain new constructions of
generalized Kähler metrics.
slides
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Divendres 15 de febrer, 15h, Aula T2, FMI-UB
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Paula Escorcielo
Universidad de Buenos Aires
Argentina
Contact:
cdandrea at ub.edu
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A version of Putinar's Positivstellensatz for cylinders
Let f be a polynomial in n variables with real coefficients.
Assume f is positive (non-negative) in a basic closed semialgebraic set S, a certificate of the positivity (non-negativity) of f in S is an expression that makes evident this fact.
For example, Hilbert's 17-th problem states that if a polynomial is non-negative in Rn, it can be written as a sum of squares of rational functions, which is a certificate of the non-negativity of f in Rn.
It is well-known that Krivine's Positivstellensatz (which states necessary and sufficient conditions for a polynomial system of equations and inequations to have no solution in Rn) implies Hilbert's 17-th problem.
There are also other versions of Positivstellensatz, which hold on particular situations.
For instance, Putinar's Positivstellensatz states that given g1, ..., gs polynomials in n variables with real coefficients such that the quadratic module M(g1, ..., gs) generated by g1, ..., gs is archimedean, every polynomial f which is positive on the basic closed semialgebraic subset S of Rn where g1, ..., gs are non-negative, belongs to M(g1, ..., gs). The archimedeanity assumption on M(g1, ..., gs) implies that the set S is compact.
In this talk, we will present a version of Putinar's Positivstellensatz in the case that the underlying basic closed semialgebraic set is not compact but a cylinder of type SxR.
This is a joint work with Daniel Perrucci.
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Dilluns 18 de febrer, 15h, aula T2, FMI-UB
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Thomas Strobl
Université Claude Bernard, Institut Camille Jordan, Lyon 1, França
Contact:
eva.miranda at upc.edu
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The universal Lie ∞-algebroid of a singular foliation
We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated 𝒪-submodule of vector fields on the underlying manifold closed under Lie bracket, where 𝒪 is the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie ∞-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie ∞-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie ∞-algebroid of the singular foliation. For real analytic or holomorphic singular foliations, it can be chosen, locally, to be a Lie n-algebroid for some finite n.
If time permits we mention how to apply this construction to the realm of geometrical invariants and/or the construction of gauge theories.
This is joint work with Camille Laurent-Gengoux and Sylvain Lavau.
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Divendres 22 de febrer, 15h, Aula T2, FMI-UB
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Yairon Cid Ruiz
UB
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Saturated special fiber ring and rational maps
The idea of studying rational maps by looking at the syzygies of the base ideal is a relatively new idea that has now become an important research topic.
In this talk, we will discuss some recent results that lead to birationality criteria and formulas for the degree of rational maps that depend on the algebraic properties of the syzygies of the base ideal.
Mainly, we will introduce a new algebra called «saturated special fiber ring» and we will discuss its relations with the degree and birationality of rational maps between irreducible projective varieties.
Time permitting, we will also discuss some results in the problem of specializing the coefficients of a rational map.
This talk is based on joint works with Laurent Busé and Carlos D’Andrea and with Aron Simis.
slides
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Divendres 1 de març, 15h, Aula T2, FMI-UB
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Ferran Dachs-Cadefau
Martin-Luther-Universität Halle, Alemanya
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Multiplicities of jumping points for mixed multiplier ideals
In this talk I want to present a systematic study of the multiplicity of the jumping points associated
to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities.
In particular we study the behaviour of the multiplicity by small perturbations of the jumping points.
We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. If time allows,
we would present some results about which topological information can be deduced from the
jumping walls.
This is a joint work with Maria Alberich-Carramiñana, Josep Àlvarez Montaner and Víctor González Alonso.
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Divendres 15 de març, 15h, Aula T2, FMI-UB
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Nick Vannieuwenhoven
Katholieke Universiteit Leuven, Bèlgica
Contact:
alessandro.oneto at upc.edu
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Geometry of the tensor rank decomposition
The tensor rank decomposition or CPD expresses a tensor as a minimum-length linear combination of
elementary rank-1 tensors.
It has found application in fields as diverse as algebraic statistics, psychometrics, chemometrics,
signal processing and machine learning, mainly for data analysis purposes.
In these applications, the theoretical model is oftentimes a low-rank CPD and
the elementary rank-1 tensors are usually the quantity of interest. However, in practice,
this mathematical model is corrupted by measurement or sampling errors.
In this talk, we will investigate the sensitivity of the CPD using techniques from algebraic
and differential geometry.
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Divendres 22 de març, 15h, Aula T2, FMI-UB
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Jarosław Buczyński
Uniwersytet Warszawski - IMPAN, Polònia
Contact:
jroe at mat.uab.cat
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Strassen's additivity of tensor rank for small three-way tensors
For a tensor \(T\in A \otimes B \otimes C\)
(for vector spaces \(A\), \(B\) and \(C\))
the tensor rank of \(T\) is the minimal number of simple tensors such that \(T\) is the sum of those simple tensors.
In this talk we address the problem of the additivity of the tensor rank.
That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks.
A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov.
The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces.
We show that for some small tensors the additivity holds.
For instance, if the rank of one of the tensors is at most 6, then the additivity holds.
Or, if one of the tensors lives in \(\mathbb{C}^k \otimes \mathbb{C}^3 \otimes \mathbb{C}^3\) for any \(k\), then the additivity also holds.
Based on a joint work with Elisa Postinghel and Filip Rupniewski.
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Divendres 22 de març, 16h, Aula T2, FMI-UB
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Patricio Almirón
Universidad Complutense de Madrid
Contact:
maria.alberich at upc.edu
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On the quotient of Milnor and Tjurina number
Two of the main invariants of plane curves singularities are Milnor number μ, of topological nature, and Tjurina number τ, of analytical nature. In 2017 A. Dimca and G.M. Greuel posed the following question:
Is it true that for any isolated plane curve singularity we have μ/τ<4/3?
In this talk I will present a partial answer to this question in the case of semi-quasihomogeneus singularities (joint work with G. Blanco) which constitute a new evidence to believe in a positive answer to Dimca and Greuel's question in the general case.
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Divendres 29 de març, 15h, Aula T2, FMI-UB
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Joan Carles Naranjo
Universitat de Barcelona
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Hyperelliptic Jacobians and Isogenies
We mainly consider abelian varieties isogenous to hyperelliptic Jacobians.
In the first part of the talk we will prove that a very general hyperelliptic Jacobian of genus \(g\geq 4\)
is not isogenous to a non-hyperelliptic Jacobian.
As a consequence we will obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian.
Another corollary is that the Jacobian of a very general \(d\)-gonal curve of genus \(g\geq 4\) is not isogenous to a different Jacobian.
In the second part we will consider a closed subvariety \(\mathcal Y \subset \mathcal A_g\) of the moduli space of principally polarized
varieties of dimension \(g\geq 4\).
We will show that if a very general element of \(\mathcal Y\) is dominated by the Jacobian of a curve \(C\) and \(\dim \mathcal Y\geq 2g\), then \(C\) is not hyperelliptic.
In particular, if the general element in \(\mathcal Y\) is simple, its Kummer variety does not contain rational curves.
Finally, if time permits, we will show that a closed subvariety \(\mathcal Y\subset \mathcal M_g\) of dimension \(2g-1\)
such that the Jacobian of a very general element of \(\mathcal Y\) is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus.
These results have been obtained in collaboration with G.P. Pirola and can be found in arXiv:1705.10154v2 or https://doi.org/10.1016/j.aim.2018.07.025. |
Divendres 5 d'abril, 15:10h, Aula T2, FMI-UB
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Federico Caucci
Università di Roma 1 La Sapienza, Itàlia
Contact:
marti.lahoz at ub.edu
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Derived invariants arising from the Albanese map
Given a smooth complex projective variety, it is natural to ask which geometric information are preserved under derived equivalence.
Namely, if two varieties have equivalent derived categories, what can we say about their geometry? We prove a general result in this direction: the derived
invariance of the cohomology ranks of pushforward under the Albanese map of the canonical line bundle (twisted with elements of the Picard variety).
In the case of maximal Albanese dimension this settles conjectures of Popa and Lombardi-Popa, including the derived invariance of the Hodge numbers \(h^{0,j}\).
This is a joint work with G. Pareschi.
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Divendres 26 d'abril, 15:10h, Aula T2, FMI-UB
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Hóa Trần Quang
IMUB and Hue University, Vietnam
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Geometry of rational maps via syzygies
The study of rational maps is of theoretical interest in algebraic geometry and commutative algebra, and of practical importance in geometric modeling.
In particular, rational maps in low dimension, typically parameterizations of curves and surfaces embedded in the projective space of dimension 3,
are applied to shape modeling using computer-aided design methods for curves and surfaces in geometric modeling.
Therefore, it is of vital importance to have a detailed knowledge of the geometry of these parametric representations.
In this talk, I will describe how geometric properties of a given rational map between projective spaces can be extracted from
the syzygies of its defining homogeneous polynomials.
Firstly, I will present our recent results on bounding the number of one-dimensional fibers of a rational map from \(\mathbb{P}^2\) to \(\mathbb{P}^3\).
Secondly, I will discuss the effective criteria of birationality for dominant rational maps from \(\mathbb{P}^1\times \mathbb{P}^1\) to \(\mathbb{P}^2\).
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Divendres 3 de maig, 15:10h, Aula T2, FMI-UB
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Marcin Dumnicki
Uniwersytet Jagielloński w Krakowie, Polònia
Contact:
jroe at mat.uab.cat
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Bound for Waldchmidt constants
I will define the Waldschmidt constant of a homogeneous ideal \(I\) to be the limit of the sequence \(a(I^{(m)})/m\), where \(a(J)\) is the least degree of a non-zero form in \(J\), and \(I^{(m)}\) is the symbolic power of \(I\). I focus on bounds on Waldschmidt constant. Bounding from below usually requires methods, which can prove emptiness of some linear systems. I will recall some general results, as well as some close bounds for points and lines in projective spaces. Bounds from above are less known, but I will introduce one new invariant (asymptotic Hilbert polynomial) and use it to find good upper bounds.
slides
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Divendres 10 de maig, 15:10h, Aula T1, FMI-UB
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Sascha Timme
Technische Universität Berlin, Alemanya
Contact:
piotr.zwiernik at upf.edu
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3264 Conics in a Second
Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. In this talk I want to illustrate how these two fields complement each other. The focus lies on the 3264 conics that are tangent to five given conics in the plane. I will illustrate tools and techniques used in numerical algebraic geometry and how we used these to find a fully real instance of this classic problem.
This is joint work with P. Breiding and B. Sturmfels.
slides
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Divendres 17 de maig, 15:10h, Aula T2, FMI-UB
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Juan Margalef
UPC
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Parametrized theories: a way towards General Relativity
What does it mean to have a degree of freedom at the boundary? In general relativity, it is important to answer this question in order to try to compute the entropy of a Black Hole. Spoiler alert: the answer is unclear. Nonetheless, we have tried to understand such question in simpler systems focusing on its geometric features. In that talk, I will take a tour through some of the theories that we have studied in order to answer this and similar questions, namely some field theories coupled to point masses, some parametrized theories and, of course, the theory of general relativity itself.
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Divendres 24 de maig, 15:10h, Aula T1, FMI-UB
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Francesco Cannas Aghedu
Università di Cagliari, Itàlia
Contact:
eva.miranda at upc.edu
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Balanced metrics on the blow-up of C² at the origin
An interesting open question in Kähler geometry is the characterization of projectively induced Kähler metrics.
An important class of projectively induced Kähler metrics are the so called balanced metrics in the sense of Donaldson S. (Scalar curvature and projective embeddings. I. J. Diff. Geom. 59, 479-522 (2001)).
This talk is a report of two works: On the balanced condition for the Eguchi-Hanson metric (J. of Geom. and Phy. 137, (2019), 35-39) and the second one joint with Andrea Loi: The Simanca metric admits a regular quantization (arXiv:1809.04431).
In particular, after a brief introduction to Balanced metrics, we see the blowing-up operation and we introduce two important complete Kähler metrics defined on the blow-up of C² at the origin: the celebrated Simanca metric and the Eguchi-Hanson metric.
The Simanca metric is an important example (both from mathematical and physical point of view) of non homogeneous complete, zero constant scalar curvature metric.
The Eguchi-Hanson metric is a well-known example of non homogeneous complete, Ricci-flat Kähler metric.
In this talk we discuss on the balanced condition for these two Kähler metrics and their consequences.
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Divendres 7 de juny, 15:10h, Aula T1, FMI-UB
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Roberto Rubio
UAB
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An introduction to generalized geometry
Generalized geometry is an approach to geometric structures pioneered by Hitchin in 2003. In this talk I will focus on generalized complex geometry, as developed by Gualtieri, in order to introduce the main ideas and definitions, some of which are borrowed from Dirac geometry, such as the Dorfman bracket, or the concept of a Courant algebroid. I will try to illustrate the threefold value of generalized geometry by encompassing various structures (symplectic and complex structures are both generalized complex structures), reinterpreting involved ones (bihermitian geometry corresponds to generalized K\"ahler geometry), and introducing genuinely new structures (there are neither symplectic nor complex manifolds admitting generalized complex structures). Time permitting I will hint at some more recent directions of the theory.
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Divendres 14 de juny, 15:10h, Aula T1, FMI-UB
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Gian Pietro Pirola
Università di Pavia, Itàlia
Contact:
jcnaranjo at ub.edu
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On the local geometry of the period map of curves
The period map \(\tau:M_g\to A_g\) gives an immersion (outside the hyperelliptic locus)
of the moduli space of complex curves of genus \(g\) into the moduli space of principally
polarized abelian varieties of dimension \(g\). We study the local geometry of this immersion
by means of the natural riemannian (orbifold) structure induced on \(A_g\) from Siegel space.
In particular two methods to give a bound on the dimension of the totally geodesic subvarieties
of \(A_g\) contained in \(M_g\) are discussed. The first one (Colombo-Frediani-Ghigi) uses
the second fundamental form associated to the Torelli immersion and the second one (Ghigi-P-Torelli)
uses instead a sort of local Fujita decomposition along geodesics.
We recall that the Shimura varieties are (algebraic) totally geodesic subvarieties of \(A_g\) and for
\(g\gg 0\), according to the Coleman-Oort conjecture, they should not be contained in \(\tau(M_g)\).
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Divendres 28 de juny, 15:10h, Aula T2, FMI-UB
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Cloenda del Seminari 2018-2019 |
Divendres 5 de juliol
Aula T1, FMI-UB |
Ciro Ciliberto
Università di Roma Tor Vergata, Itàlia
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Nodal curves on Enriques surfaces
In this talk I will consider Severi varieties of nodal curves in linear systems on Enriques surfaces. The problems I will treat are: the dimensionality problem of components of the Severi variety and the existence problem of such components on a general Enriques surface.
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15:10h
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Cafè |
16:10h
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Guillem Blanco
Universitat Politècnica de Catalunya
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Yano's conjectureIn 1982, T. Yano proposed a conjecture about the generic \( b \)-exponents of an irreducible plane curve singularity. Given any holomorphic function \( f : (\mathbb{C}^2, \boldsymbol{0}) \longrightarrow (\mathbb{C}, 0) \) defining an irreducible plane curve, the conjecture gives an explicit formula for the generic \( b \)-exponents of the singularity in terms of the characteristic sequence of \( f \). In this talk, we will present a proof of Yano's conjecture.
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16:40h
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Reunió - organització seminari 2019-2020 |
17:40h
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Rick Miranda
Colorado State University, EUA
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Nonlinear Toric Degenerations
We are motivated by considerations related to the Segre-Gimigliano-Harbourne-Hirschowitz conjecture, which deals with the dimension of linear systems of plane curves with prescribed multiple points. The plane is a toric surface, and several approaches to the SGHH problem involve toric constructions leading to auxiliary questions. One is: when does one toric surface degenerate to another? I’ll present several examples and a general technique which hopefully illustrate some of the subject’s general interest. Most of the talk should be accessible to non-experts.
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18:15h
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Sopar |
20:30
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